Prove this inequality: b + c a + c + a b + a + b c ≥ a + b + c + 3I tried various methods. But, couldn't solve it. It'd be great if anyone can help.

Question

Prove this inequality:            b      +      c              a        +            c      +      a              b        +            a      +      b              c        ≥      a    +      b    +      c    +  3I tried various methods. But, couldn&#039;t solve it. It&#039;d be great if anyone can help.

Syllingbs · Accepted Answer

By AM-GM            b      +      c              a        +            c      +      a              b        +            a      +      b              c        ≥            2              b        c                    a        +            c      a              b        +            2              a        b                    c        =      2    a    +      2    b    +      2    c  For the last equality   a  b  c  =  1 was used.Then again by AM-GM;      1    a    +      1    b    +      1    c    ≥      3                  a        b        c            3        =  3and again by a well known inequality      1    a    +      1    b    +      1    c    ≥      1          b      c        +      1          c      a        +      1          a      b        =      a    +      b    +      c  For the last equality   a  b  c  =  1 was used. Combine the last two inequalities to get the desired result.

raulgallerjv · Accepted Answer

You already have a proof using AM-GM and Muirhead. For a pure AM-GM proof, note that       1    2        (          a      b        +          b      a        )    ⩾  1 and so on, so half the LHS takes care of the 3 on RHS.For the rest, note you can cyclically sum the (weighted) AM-GMs:      5    18        (          a      b        +          a      c        )    +      2    18        (          b      a        +          b      c        )    +      2    18        (          c      a        +          c      b        )    ⩾            a      3                      b        c            6        =      a

Prove this inequality: b + c </mrow> <msqrt> a

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